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==Statement==
Let {''X<sub>n</sub>''}, ''X'' be [[random element]]s defined on a [[metric space]] ''S''. Suppose a function ''g'': ''S''→''S′'' (where ''S′'' is another metric space) has the set of [[Discontinuity (mathematics)|discontinuity points]] ''D<sub>g</sub>'' such that Pr[''X''∈''D<sub>g</sub>''] = 0. Then <ref>{{harvnb|van der Vaart|1998|loc=Theorem 2.3}}</ref><ref>{{harvnb|Billingsley|1969|page=31, Corollary 1}}</ref><ref>{{harvnb|Billingsley|1999|page=21, Theorem 2.7}}</ref>
# <math>X_n \ \xrightarrow{d}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{d}\ g(X);</math>
# <math>X_n \ \xrightarrow{p}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{p}\ g(X);</math>
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